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Theorem isusp 23766
Description: The predicate 𝑊 is a uniform space. (Contributed by Thierry Arnoux, 4-Dec-2017.)
Hypotheses
Ref Expression
isusp.1 𝐵 = (Base‘𝑊)
isusp.2 𝑈 = (UnifSt‘𝑊)
isusp.3 𝐽 = (TopOpen‘𝑊)
Assertion
Ref Expression
isusp (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)))

Proof of Theorem isusp
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elex 3493 . 2 (𝑊 ∈ UnifSp → 𝑊 ∈ V)
2 0nep0 5357 . . . . 5 ∅ ≠ {∅}
3 isusp.1 . . . . . . . . . . . 12 𝐵 = (Base‘𝑊)
4 fvprc 6884 . . . . . . . . . . . 12 𝑊 ∈ V → (Base‘𝑊) = ∅)
53, 4eqtrid 2785 . . . . . . . . . . 11 𝑊 ∈ V → 𝐵 = ∅)
65fveq2d 6896 . . . . . . . . . 10 𝑊 ∈ V → (UnifOn‘𝐵) = (UnifOn‘∅))
7 ust0 23724 . . . . . . . . . 10 (UnifOn‘∅) = {{∅}}
86, 7eqtrdi 2789 . . . . . . . . 9 𝑊 ∈ V → (UnifOn‘𝐵) = {{∅}})
98eleq2d 2820 . . . . . . . 8 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ 𝑈 ∈ {{∅}}))
10 isusp.2 . . . . . . . . . 10 𝑈 = (UnifSt‘𝑊)
1110fvexi 6906 . . . . . . . . 9 𝑈 ∈ V
1211elsn 4644 . . . . . . . 8 (𝑈 ∈ {{∅}} ↔ 𝑈 = {∅})
139, 12bitrdi 287 . . . . . . 7 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ 𝑈 = {∅}))
14 fvprc 6884 . . . . . . . . 9 𝑊 ∈ V → (UnifSt‘𝑊) = ∅)
1510, 14eqtrid 2785 . . . . . . . 8 𝑊 ∈ V → 𝑈 = ∅)
1615eqeq1d 2735 . . . . . . 7 𝑊 ∈ V → (𝑈 = {∅} ↔ ∅ = {∅}))
1713, 16bitrd 279 . . . . . 6 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ ∅ = {∅}))
1817necon3bbid 2979 . . . . 5 𝑊 ∈ V → (¬ 𝑈 ∈ (UnifOn‘𝐵) ↔ ∅ ≠ {∅}))
192, 18mpbiri 258 . . . 4 𝑊 ∈ V → ¬ 𝑈 ∈ (UnifOn‘𝐵))
2019con4i 114 . . 3 (𝑈 ∈ (UnifOn‘𝐵) → 𝑊 ∈ V)
2120adantr 482 . 2 ((𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)) → 𝑊 ∈ V)
22 fveq2 6892 . . . . . 6 (𝑤 = 𝑊 → (UnifSt‘𝑤) = (UnifSt‘𝑊))
2322, 10eqtr4di 2791 . . . . 5 (𝑤 = 𝑊 → (UnifSt‘𝑤) = 𝑈)
24 fveq2 6892 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
2524, 3eqtr4di 2791 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
2625fveq2d 6896 . . . . 5 (𝑤 = 𝑊 → (UnifOn‘(Base‘𝑤)) = (UnifOn‘𝐵))
2723, 26eleq12d 2828 . . . 4 (𝑤 = 𝑊 → ((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ↔ 𝑈 ∈ (UnifOn‘𝐵)))
28 fveq2 6892 . . . . . 6 (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
29 isusp.3 . . . . . 6 𝐽 = (TopOpen‘𝑊)
3028, 29eqtr4di 2791 . . . . 5 (𝑤 = 𝑊 → (TopOpen‘𝑤) = 𝐽)
3123fveq2d 6896 . . . . 5 (𝑤 = 𝑊 → (unifTop‘(UnifSt‘𝑤)) = (unifTop‘𝑈))
3230, 31eqeq12d 2749 . . . 4 (𝑤 = 𝑊 → ((TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤)) ↔ 𝐽 = (unifTop‘𝑈)))
3327, 32anbi12d 632 . . 3 (𝑤 = 𝑊 → (((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ∧ (TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤))) ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈))))
34 df-usp 23762 . . 3 UnifSp = {𝑤 ∣ ((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ∧ (TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤)))}
3533, 34elab2g 3671 . 2 (𝑊 ∈ V → (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈))))
361, 21, 35pm5.21nii 380 1 (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2941  Vcvv 3475  c0 4323  {csn 4629  cfv 6544  Basecbs 17144  TopOpenctopn 17367  UnifOncust 23704  unifTopcutop 23735  UnifStcuss 23758  UnifSpcusp 23759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-iota 6496  df-fun 6546  df-fv 6552  df-ust 23705  df-usp 23762
This theorem is referenced by:  ressust  23768  ressusp  23769  tususp  23777  uspreg  23779  ucncn  23790  neipcfilu  23801  ucnextcn  23809  xmsusp  24078
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